you should have caught yourself by doing it again. to get rid of the $\displaystyle - \frac 12$ in front, you multiplied through by -2, correct? what is $\displaystyle -2 \left( 2tu\right)$?

EDIT: Ok, you caught yourself, good job. well, it wasn't a waste of time. you'll be more careful with such things from now on, and it will save you time in the future. we tend to make less mistakes if it hurts to make them

you made another mistake here. you inverted both sides incorrectly. combine the fractions on the right first, then flip both sides. (by the way, i would multiply through by $\displaystyle t^4$ just before that)

Same question applies whether I need +/- or not. I'm hoping this is right now.

yes, we need the +/- here. we have no basis on which to choose the positive over the negative. if we were given initial conditions or some other conditions to fulfill that would cause a negative or a positive answer to be impossible, then we would drop one of the signs